An orbital is a mathematical solution to the Schrödinger equation. These solutions describe where an electron is most likely to be in an atom. But can we know exactly where the electron is? No. What we can do is calculate the probability of finding it in a certain region, based on its wavefunction.

Wave Behaviour and Energy Levels

Electrons don’t move in fixed paths like planets around the sun. They behave more like waves. Inside an atom, these waves are standing waves, formed because the electron is confined by electrostatic forces. These standing wave patterns give orbitals their distinct shapes and sizes.

Because of their wave-like nature, electrons can’t have just any energy. Their energy is quantized, meaning they’re restricted to certain values that satisfy the Schrödinger equation. Each allowed energy corresponds to an orbital, which is defined by a specific set of quantum numbers.

Quantum Numbers and Orbital Structure

Check out my post on quantum numbers for more detail. But as a quick recap, the principal quantum number (n) tells us the size and energy of the orbital. Higher values of n mean larger and more energetic orbitals. The angular momentum quantum number (l) determines the shape of the orbital (whether it’s spherical, dumbbell-shaped, or something more complex). Finally, the magnetic quantum number (mₗ) describes how the orbital is oriented in space.

In multi-electron atoms, energy depends not just on n, but also on l. That’s why orbitals like 2s and 2p, even though they share the same n, have different energies. Interactions between electrons and shielding effects cause these shifts, which we’ll look at more closely soon.

Orbital Phase and Bonding

To understand how atoms bond, it helps to understand orbital phase. Just like waves have crests and troughs, orbital wavefunctions have positive and negative regions. When orbitals overlap, their phases matter. If the phases align, the overlap is constructive and leads to bonding. If they clash, they cancel each other out, resulting in antibonding interactions.

Shapes of Atomic Orbitals

The shapes of orbitals emerge directly from the quantum numbers. s orbitals are spherical, symmetrical in all directions. 

p orbitals look like dumbbells aligned along one of the three axes: x, y, or z. Each has a nodal plane slicing through the nucleus where the electron can’t be found, dividing the orbital into two lobes.

d orbitals are more complex. There are five in total. Three of them (d(xy), d(yz), d(xz)) look like four-leaf clovers, with lobes pointing between the axes. The d(𝓏²) orbital is shaped like a dumbbell with a donut around its middle. The dₓ²–ᵧ² orbital stretches its lobes along the x and y axes.

f orbitals are populated in heavier elements like the lanthanides and actinides. They’re even more intricate, with three angular nodes and shapes that mix lobes and donut-like regions.

g orbitals, associated with l = 4, aren’t populated in ground-state atoms we know, but they’re part of the quantum model. They’d be expected in very high-energy atoms with complex shapes and four angular nodes.

What Are Nodes?

Nodes are regions where the probability of finding an electron is zero. There are two types. Radial nodes are like hollow spheres centered on the nucleus. Angular nodes are planes or cones slicing through the orbital.

The total number of nodes in any orbital is n – 1. The number of angular nodes depends on l. So, for example, a p orbital (l = 1) has one angular node, and the rest of its nodes (if any) are radial.

Multi-Electron Atoms

In multi-electron atoms, the energy of an orbital depends on both n and l, and two main effects (shielding and electron-electron repulsion) are responsible.

Shielding happens when inner electrons block the outer ones from feeling the full charge of the nucleus. How much shielding occurs depends on how much the orbital penetrates toward the nucleus. s orbitals, for example, can get closer in than p or d orbitals, so they feel a stronger nuclear pull and are lower in energy.

That’s why in the third shell, the 3s orbital is lower in energy than 3p, and 3p is lower than 3d.

Repulsion between electrons is the second factor. As more electrons are packed into an atom, they push against each other. The shapes of orbitals, especially complex ones like the 3d set, help reduce this repulsion.